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Multi-Compartment Model of Brain Tissues from T2Relaxometry MRI Using Gamma Distribution

Sudhanya Chatterjee, Olivier Commowick, Onur Afacan, Simon K Warfield,Christian Barillot

To cite this version:Sudhanya Chatterjee, Olivier Commowick, Onur Afacan, Simon K Warfield, Christian Barillot. Multi-Compartment Model of Brain Tissues from T2 Relaxometry MRI Using Gamma Distribution. ISBI2018 - IEEE International Symposium on Biomedical Imaging, Apr 2018, Washington DC, UnitedStates. pp.141-144, �10.1109/ISBI.2018.8363541�. �hal-01744852�

MULTI-COMPARTMENT MODEL OF BRAIN TISSUES FROM T2 RELAXOMETRY MRIUSING GAMMA DISTRIBUTION

Sudhanya Chatterjee? Olivier Commowick? Onur Afacan† Simon K. Warfield† Christian Barillot?

?VisAGeS U1228 INSERM/Inria, IRISA UMR CNRS 6074, University of Rennes 1, France†Computational Radiology Laboratory, Boston Children’s Hospital, Boston, MA, USA

ABSTRACT

The brain microstructure, especially myelinated axons andfree fluids, may provide useful insight into brain neurodegen-erative diseases such as multiple sclerosis (MS). These maybe distinguished based on their transverse relaxation timeswhich can be measured using T2 relaxometry MRI. However,due to physical limitations on achievable resolution, eachvoxel contains a combination of these tissues, rendering theestimation complex. We present a novel multi-compartmentT2 (MCT2) estimation based on variable projection, appli-cable to any MCT2 microstructure model. We derive thisestimation for a three-gamma distribution model. We validateour framework on synthetic data and illustrate its potential onhealthy volunteer and MS patient data.

Index Terms— T2 relaxometry, microstructure, brain

1. INTRODUCTION

MRI voxels of the human brain are heterogeneous in terms oftissue types due to the limited imaging resolution and physicalconstraints. Each voxel in the white matter (WM) contains alarge number of myelinated and non-myelinated axons, glialcells and extracellular fluids [1, 2]. For example, every squaremillimeter of the corpus callosum in a human brain has morethan 100,000 fibers (myelinated and non-myelinated) of vary-ing diameters [1]. These tissues can be distinguished basedon their T2 relaxation times. Myelin being a tightly wrappedstructure has a very short T2 relaxation time of 10 millisec-onds (ms) [2]. The estimated T2 relaxation time of the myeli-nated axons is 40ms[2]. The ventricles and tissue injuryregions contain free fluids which have a high T2 relaxationtime (>1000ms). The T2 relaxation values between those ofmyelin and myelinated axons and the free fluids correspondto the glial cells and extracellular tissues [2]. An ability toobtain the condition of these tissues can help us gain betterinsights into the onset and progress of neurodegenerative dis-eases such as multiple sclerosis (MS).Myelin water fraction (MWF) has been computed from T2 re-laxometry images using a variety of approaches [3, 4]. Mostof these methods primarily focus on the MWF estimation.However, MWF alone might not be able to convey the en-

tire information since it is a relative measurement. For ex-ample, in MS patients a decrease in MWF in a WM lesionmight be caused by myelin loss or fluid accumulation due totissue injury or both. Hence for relative measurements likewater fraction (WF), all the WF maps should be observed si-multaneously for a complete understanding of the tissue con-dition. Here we propose an estimation framework to obtainbrain microstructure information using a multi-compartmenttissue model from T2 relaxometry MRI data. The T2 spaceis modeled as a weighted mixture of three continuous prob-ability density functions (PDF), each representing the tissueswith short, medium and high T2 relaxation times. We estimatethe PDF parameters using variable projection (VARPRO) ap-proach [5]. We derive this generic estimation framework forgamma PDFs. We validate the proposed method using syn-thetic data against known ground truth. We then illustrated iton a healthy subject and MS patient.

2. METHOD AND MATERIALS

2.1. Theory

Signal model The T2 space is modeled as a weighted mix-ture of three PDFs, representing each of the three T2 relaxom-etry compartments: short−, medium− and high−T2. Thusthe voxel signal at the i−th echo time (ti) is given as:

s (ti) = M0

3∑j=1

wj

∞∫0

fj (T2; pj) EPG (T2,4TE, i, B1) dT2

(1)Each compartment is described by a chosen PDF, fj (T2; pj),where pj = {pj1 , . . . , pjn} ∈ Rn are the PDF parame-ters. In Eq. (1), wj is the weight of the j-th distributionwith

∑j wj = 1. ∆TE, B1 and M0 are the echo spacing,

field inhomogeneity and magnetization constant respectively.Imperfect rephasing of the nuclear spins after application ofrefocusing pulses leads to the generation of stimulated echoes[6]. Hence the T2 decay is not purely exponential. The stim-ulated echoes are thus obtained using the EPG algorithm[7]. EPG(·) is the stimulated echo computed at ti = i∆TEwhere i = {1, . . . ,m} and m is the number of echoes.

Optimization M0 and wj can be combined into a singleterm αj ∈ R+ without any loss of generality. In that case, theweight wj corresponding to each compartment is obtained aswj = αj/

∑i αi. In the most general case, the least squares

minimization problem is thus formulated as:

(α, p, B1

)= arg min

α,p,B1

m∑i=1

yi − 3∑j=1

αjλj (ti; p, B1)

2

= arg minα,p,B1

‖Y −Λ (p, B1)α‖22 (2)

where Y ∈ Rm is the observed signal and m is the numberof echoes; α ∈ R+3

; Λ ∈ Rm×3; p = {p1,p2,p3} ∈Rk, where k = 3n. In Eq. (2), each element of Λ,Λij =λ (ti; pj, B1), is computed as:

Λij =

∞∫0

fj (T2; pj)EPG (T2,4TE, i, B1) dT2 (3)

Due to the EPG formulation, there is no closed form deriva-tive solution for the optimization ofB1 [7]. Hence, we opt foran alternate optimization scheme where we iterate betweenoptimization of {p, α} with a fixed value of B1 and opti-mization for B1 using the obtained {p, α} values. The termsΛ (p, B1) and α in Eq. (2) are linearly separable. Hence wecan use the VARPRO approach to solve for {p, α} [5]. Theunknown α is substituted by Λ (p)

+Y, where Λ (p)

+ is theMoore-Penrose generalized inverse of Λ (p). The VARPROcost function is computed as:

arg minp

∥∥∥(I−Λ (p) Λ (p)+)

Y∥∥∥22

(4)

where, I − Λ (p) Λ (p)+ is the projector on the orthogonal

complement of the column space of Λ (p). Since p ∈ Rk,the Jacobian matrix J ∈ Rk×m and its columns are computedas shown in [5]. To compute the elements of J, we need toobtain ∂Λ/∂pji , ∀i, j [5]. After solving Eq. (4) for p, thevalues of α are obtained as Λ (p)

+Y. The optimization for

{α,p} and B1 is performed alternatively until convergence.B1 is optimized using a gradient free optimizer (BOBYQA),as it does not have any closed form solution [7].Multi-compartment model using gamma PDF The previ-ous estimation framework is generic as it does not depend onthe chosen PDF. We choose here to use gamma PDF for fj(·)for j = {1, 2 ,3} since their non-negativity and skewed natureare well suited to describe the compartments used to modelthe T2 space. The mean T2 values of myelin, myelinated ax-ons, inter- and extra-cellular and free fluids in the brain arewell studied in the literature [2, 3]. Hence we parameterizedeach fj in terms of its mean (µj) and variance (vj) rather thanthe usual shape and scale parameter representation (refer Eq.(5)). Using this parametric form of the gamma PDF makes

the choice of optimization bounds convenient.

f (T2;µj , vj) =T

(µ2j/vj)−1

2

Γ(µ2j/vj

)(vj/µj)

µ2jvj

exp

(−T2vj/µj

)(5)

Hence we have, p = {µs , vs, µm, vm, µh, vh} where, (·)s,(·)m and (·)h are the PDF parameters describing the short-, medium- and high-T2 compartments respectively. Due topractical limitations such as feasible acquisition time, coilheating and specific absorption rate (SAR) guidelines, T2 re-laxometry MRI sequences have limitations on the shortestecho time and number of echoes per acquisition. The high-T2 compartment aims at capturing of free fluids in the brain,and hence has a T2 relaxation time larger than 1 second [3]. Astandard T2 spin echo multi-contrast sequence has the shortestT2 acquisition (first echo) at around 8−10ms and has 20−40acquired echoes. Hence for the short-T2 compartment, weusually have a very limited (around 3-4) number of echoes.There are almost no echoes available which correspond to thehigh-T2 compartment. The robustness and accuracy of theimplementations to simultaneously estimate the weights andall the PDF parameters has been found to be not reliable [8].Hence we choose to estimate only the mean of the gammaPDF corresponding to the medium-T2 compartment.Using the VARPRO approach we thus estimate four param-eters of the signal model: mean of the medium-T2 gammaPDF (µm) and the three weights corresponding to each com-partment. Hence only ∂Λ/∂µm is required for computing theJacobian matrix and is obtained as:

∂Λ

∂µm=

∞∫0

f (T2;µm, vm)

[µmvm

(2 log

(T2

µmvm

)− (6)

2Ψ

(µ2m

vm

)+ 1

)− T2vm

]EPG (T2,4TE, i, B1) dT2

where Ψ(·) is the digamma function. The remaining gammaPDF parameter values are pre-selected for the three compart-ments based on histology findings reported in the literature [3]and are set as {µs, µh} = {30, 2000}ms and {vs, vm, vh} ={50, 100, 6400} ms2. We assume a reasonable bound on µmof 100-125ms for its optimization. The minimization prob-lem in Eq. (4) is solved for µm using the analytically obtainedderivative in Eq. (6) with a gradient based optimizer [9]. Theshort-T2 compartment here indicates the condition of myelinand myelinated axons [2]. The medium-T2 compartment’sWF conveys information on the condition of axons, glial cellsand extracellular fluids [2]. The condition of free fluids (suchas in ventricles and fluid accumulation due to tissue injuries)is indicated by the high-T2 WF values.

2.2. Experiments

Synthetic data. The proposed method was first validatedagainst synthetic data generated following a known ground

truth, composed of three gamma PDFs with parameters, {µs,µm, µh} = {25, 120, 1900} and {vs, vm, vh} = {40, 90,6000}. The weights chosen for each compartment were, {ws,wm, wh} = {0.2, 0.7, 0.1}. The B1, M0 and T1 valuesconsidered for this simulation were 1.3, 950 and 1000 re-spectively. The experiments are carried out for SNR valuesranging from 5 to 100 in steps of 5. We simulated T2 relax-ometry data with the following parameters: first echo (TE0)is at 9ms;4TE= 9ms; 32 echoes, 100 signal averages.Healthy volunteer data. The method was tested on T2relaxometry data acquired on a healthy volunteer (male,age: 26) with the following acquisition details: Siemens 3TMRI scanner; 2D multislice CPMG sequence; 32 echoes;TE0= 9ms; echo spacing of 9ms; TR = 3000ms; singleslice acquired; slice thickness of 4mm; in plane resolution of1.04mm× 1.04mm; matrix size of 192× 192.MS patient data. The method was finally tested on T2 re-laxometry MRI data of a MS patient. The observations fromour estimation maps were compared with the pathologicalfindings on MS lesion reported in the literature [10, 11]. Weobserved whether the estimation maps obtained from ourmethod were able to provide insight into MS lesion whichcorroborate with the pathological findings. The acquisitiondetails are the same as for the healthy volunteer data.

3. RESULTS

Synthetic data. The results of the synthetic data simulationare shown in Fig 1. It shows that with increasing SNR theweights estimation gets more accurate. The bars around themean value are the 95% confidence intervals (CI) which isobtained as 1.96 times the standard deviation of the estimates.The CIs of the estimation improve with with increasing SNRfor all three WFs. The ground truth lies in the CI of the meanestimated weights for all three compartments.

Fig. 1. Mean of estimated weights with a 95% confidenceinterval for 100 signal averages for the synthetic data study.

Healthy volunteer data. The estimation maps for the healthyvolunteer are shown in Fig 2. The genu of the corpus callo-sum (CC) has higher ws values compared to any other region

in the brain. The ventricles and other regions with free fluidshave a higher µm compared to the normal appearing whitematter (NAWM) tissues. This is relevant as free fluids have ahigher T2 value compared to the relatively tightly bound tis-sues present in NAWM.MS patient data. Two lesions are present in the MR imageof the MS patient shown in Fig. 3 that are marked with redand blue arrows. We observe absence of short-T2 WF in bothlesions. The lesions and their neighboring tissues have higherwm values than NAWM tissues. The wh map shows fluidaccumulation in lesion-2 but not in lesion-1 . The estimatedmedium-T2 gamma PDF map shows a higher PDF mean forboth lesions compared to the NAWM. In lesion-1 the esti-mated µm increases (with respect to the neighboring NAWM)as it approaches the core of the lesion, but is less than the µmestimated at the ventricles where there is free fluid.

4. DISCUSSION

Our method was successfully validated against synthetic datawith known ground truth for all SNR values (refer Fig. 1).High ws values in the genu of the CC of healthy volunteerdata (refer Fig. 2) is due to the high myelin and myelinatedfibers density in this region compared to any other part of thebrain [1]. In the estimation maps of the MS patient (referFig. 3), the absence of short-T2 WF in the lesions and in itsneighboring regions can be explained by demyelination of thenerve fibers caused by MS [10][3]. Demyelination at the on-set of MS is followed by macrophage intervention leading toan increased cellular activity in the MS lesion regions [10]whose T2 relaxation time is greater than myelin and myeli-nated axons but less than those of free fluids [2]. This phe-nomenon might explain the high wm values in the lesions andin neighboring regions. Demyelination is followed by pro-gressive axonal damage and fluid accumulation (due to tissueinjuries) in MS lesions [10]. The extent of axonal damage andfluid accumulation in the lesions can provide useful informa-tion regarding the lesion state and its response to a treatment.Lesion-2 has fluid accumulation unlike lesion-1 , possibly in-dicating that the two lesions are in different stages. The con-tinuous axonal damage in the MS lesions [10][11] explainsthe higher µm in the lesion regions compared to the neighbor-ing NAWM as a higher µm value indicates tissues with lesstightly bound water. The increment in µm values in lesion-1 as we approach the lesion core from the lesion boundarymight also indicate a reduction in axon density. This is in ac-cordance with the pathology of MS lesion evolution [10][11].

5. CONCLUSION

We proposed a generic estimation method to obtain estimatesof tissue microstructure in brain by modeling the T2 spectrumas a weighted mixture of three gamma PDFs. The maps es-timated from our method can be effective in understanding

Fig. 2. Estimation maps for a healthy volunteer.

Fig. 3. Estimation maps for MS patient data. Lesion-1 and lesion-2 are marked with red and blue arrows respectively.

the heterogeneity of lesions [10] in MS patients and be usedas potential biomarkers to have information on the MS lesiongrowth stage. As a part of the future study we intend to val-idate the observations by applying the proposed estimationframework on more healthy controls and MS patient datasets.

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